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Optimal truss and frame design from projected homogenization-based topologyoptimization

Larsen, S. D.; Sigmund, O.; Groen, J. P.

Published in:Structural and Multidisciplinary Optimization

Link to article, DOI:10.1007/s00158-018-1948-9

Publication date:2018

Document VersionPeer reviewed version

Link back to DTU Orbit

Citation (APA):Larsen, S. D., Sigmund, O., & Groen, J. P. (2018). Optimal truss and frame design from projectedhomogenization-based topology optimization. Structural and Multidisciplinary Optimization, 57(4), 1461-1474.https://doi.org/10.1007/s00158-018-1948-9

Noname manuscript No.(will be inserted by the editor)

Optimal truss and frame design from projectedhomogenization-based topology optimization

S. D. Larsen · O. Sigmund · J. P. Groen

Received: date / Accepted: date

Abstract In this article, we propose a novel method to

obtain a near-optimal frame structure, based on the so-

lution of a homogenization-based topology optimization

model. The presented approach exploits the equivalence

between Michell’s problem of least-weight trusses and a

compliance minimization problem using optimal rank-

2 laminates in the low volume fraction limit. In a fully

automated procedure, a discrete structure is extracted

from the homogenization-based continuum model. This

near-optimal structure is post-optimized as a frame,

where the bending stiffness is continuously decreased,

to allow for a final design that resembles a truss struc-

ture. Numerical experiments show excellent behavior of

the method, where the final designs are close to analyt-

ical optima, and obtained in less than 10 minutes, for

various levels of detail, on a standard PC.

Keywords Optimal frame design · Optimal truss

design · Michell theory · Topology optimization

1 Introduction

A classical topic within structural optimization is to

find solutions for Michell’s problem of least-weight

trusses (Michell, 1904). Computational methods to

solve these problems of optimal truss design date back

to the early sixties, when Dorn et al. (1964) introduced

the ground structure approach. This approach requires

a fixed set of nodal joints and elements, which make

up the ground structure. The cross-sectional areas of

S. D. Larsen · O. Sigmund · J. P. GroenDepartment of Mechanical Engineering, Solid Mechanics,Technical University of Denmark, 2800 Kgs, Lyngby, Den-markTel.: +45-45254252E-mail: jergro@mek.dtu.dk

these elements are then optimized, classically using lin-

ear programming methods. In recent years two easy-to-

use implementations have been presented. Soko l (2011)

has published a 99 line code programmed in Mathemat-

ica, while Zegard and Paulino (2014) present a frame

work for arbitrary 2D domains in MATLAB.

A downside of ground structure approaches is that

the location of the nodal joints has a large influence on

the performance of the design. To get a near-optimal

solution a large set of nodes and potential elements

have to be considered. Furthermore, the large number

of members and the fact that some of these members

are overlapping poses a limit on the manufacturability

of these designs.

It is also possible to include the location of the nodes

as design variables, i.e. both size and geometry are op-

timized, as introduced by Dobbs and Felton (1969) and

Pedersen (1969). In this case a small set of nodes and

elements suffices to get a near-optimal design; however,

due to the non-linearity of the combined size and geom-

etry optimization problem the initial position of nodes

and connectivity still has a large influence on the result.

To get close to the optimal distribution of nodes

and elements, growth methods have been considered

(Rule, 1994). Martınez et al. (2007) introduce an effi-

cient growth method where sequentially a node and el-

ements are introduced, size and topology optimization

are performed, and geometry optimization is applied.

The resulting designs are close in performance to an-

alytical solutions for Michell’s problem of least-weight

trusses and obtained in a relatively short time. How-

ever, due to the nature of the heuristics involved in

the growth method, it may not always converge to a

near-optimal solution as is discussed by He and Gilbert

(2015). Another downside of the growth method is that

the procedure of finding an appropriate position to in-

2 S. D. Larsen et al.

sert a new joint becomes increasingly slow when more

members are considered.

In an approach somewhat related to the present,

He and Gilbert (2015) make use of an efficient ground

structure method, in which not all members are con-

sidered initially, but adaptively inserted (Gilbert and

Tyas, 2003). The position of this set of nodes is then

optimized in a subsequent geometry optimization step.

Furthermore, crossing elements are treated by insert-

ing new nodes at crossings, leading to near-optimal and

manufacturable designs in a short time.

In a different approach Zhou and Li (2008, 2011)

use truss-like continua to get a near-optimal distribu-

tion of material and orientations. In a semi-automated

approach starting points for ray tracing are manually

selected, the grid formed by these rays is interpreted

as a truss structure, on which subsequent size and ge-

ometry optimization is performed. Similar to this ap-

proach, Gao et al. (2017) obtain an initial ground struc-

ture for size optimization, using principal stress trajec-

tories. These trajectories are obtained when the domain

is modeled as an isotropic medium; however, here it

should be mentioned that these principal stress lines

do not necessarily correspond to the principal direc-

tions for an orthotropic truss-like material. In both of

the above mentioned approaches the initial member ar-

eas are not chosen based on the continuum model, but

are found in a subsequent sizing optimization model.

In this article, we propose a novel method to ob-

tain a near-optimal set of nodes and elements, based on

the solution of a homogenization-based topology opti-

mization model. The approach is fully automatic and

the extracted structure remains close to optimal. In a

final step, the nodal positions and element areas are

further optimized, and redundant nodes and elements

eliminated. An overview of the proposed method can

be seen in Figure 1.

As discussed by Bendsøe and Haber (1993), the

compliance minimization problem in the low volume

fraction limit, using optimal rank-2 laminates, reduces

to Michell’s problem of least-weight trusses. In this limit

a rank-2 laminate can be represented by a combination

of two orthogonal rank-1 laminates acting on the same

length-scale (Bourdin and Kohn, 2008), similar to the

truss-like continua used by Zhou and Li (2008). Such

a microstructure can be projected on a fine scale using

the method proposed in Pantz and Trabelsi (2008) and

Groen and Sigmund (2017). However, we can also use

the information of the mapping functions, required to

do the projection, to establish nodes and initial con-

nectivity. Furthermore, based on the optimal material

distribution of the continuum model, a near-optimal

initial starting guess for element areas is obtained.

?

2L

LF

Design domain and boundary conditions

Homogenization-based topology

Extracted structure

Design after size and shape optimization

Fig. 1: Proposed procedure to obtain a solution to

Michell’s problem of least-weight trusses for a Michell

cantilever

Although the projected structure is close to the op-

timal solution, it is not always in the space of statically

determinate structures. Hence, when modeled as a truss

Optimal truss and frame design from projected homogenization-based topology optimization 3

structure, the stiffness matrix can be singular. To tackle

the problem of a singular stiffness matrix in the context

of topology optimization several approaches have been

proposed. Bruns (2006) uses a pseudo-inverse method,

Washizawa et al. (2004) use Krylov subspace methods,

and Ramos Jr. and Paulino (2016) use a potential en-

ergy approach with Tikhonov regulation. However, as

a simple and reliable alternative we choose to model

the structure as a thin frame structure and gradually

decrease the bending contribution using a continuation

scheme. As an added benefit, our approach allows to

study the relation between optimized frame and truss

structures.

The combined procedure of obtaining a near-

optimal initial structure and post-optimization, for de-

signs with several hundreds of nodes, requires less than

10 minutes, using a single processor MATLAB code on a

standard PC. This short time potentially allows design-

ers to use topology optimization as an interactive tool

in the design process, (Aage et al., 2013). To demon-

strate the performance of the proposed approach five

examples are considered. The cantilever beam shown

in Figure 1, and the MBB-beam, MBB-beam with void

domain, L-shaped domain and Michell cantilever with

circular support shown in Figure 2(a-d), respectively.

The paper is organized as follows: The methodology of

the homogenization-based topology optimization is in-

troduced in Section 2. In Section 3 the theory used to

obtain a near-optimal frame structure from projection

of the rank-2 laminates is explained. The procedure to

do further optimization is introduced in Section 4. The

corresponding tests on the performance and efficiency

of the developed method are shown in Section 5. Fi-

nally, the most important conclusions of this study are

summarized in Section 6.

2 Homogenization-based topology optimization

Two orthogonal rank-1 laminates are used as microstruc-

ture to perform the homogenization-based topology op-

timization. The microstructures are defined by µ1, and

µ2, which are the relative widths of the isotropic ma-

terial in layer 1 and 2 respectively, and angle θ, which

describes the angle between the material frame of refer-

ence to the global frame of reference. The correspond-

ing constitutive properties C in the global coordinate

system are written in matrix form using Voigt notation;

furthermore, the macroscopic volume fraction m can be

calculated as,

C(µ1, µ2, θ) = RT (θ)

Eµ1 0 0

0 Eµ2 0

0 0 0

R(θ)

+Emin1− ν2

1 ν 0

ν 1 0

0 0 1−ν2

m(µ1, µ2) = µ1 + µ2

(1)

Here E is the Young’s modulus of the isotropic material,

and R is the transformation matrix rotating constitu-

tive properties from the material frame of reference to

the global frame of reference. For stability reasons a

small isotropic stiffness (Emin = 0.005E and ν = 0.3)

is added to the composite constitutive properties.

Design domain Ω is discretized in ne bi-linear fi-

nite elements, each consisting of a uniform microstruc-

ture described by local design variables µ1, µ2, and θ.

The optimization problem, aimed at minimizing com-

pliance J , is solved in a nested-approach, where the de-

sign vectors describing the relative widths µ1, and µ2

are updated using the Method of Moving Asymptotes

(MMA), (Svanberg, 1987). As discussed by Pedersen

(1989, 1990), the optimal orientation of an orthotropic

composite is along the directions of principal stresses,

hence at each design iteration the angles θ are updated

accordingly. In Michell’s problem of least-weight trusses

the relative material distribution has to be found, hence

µ1 and µ2 are only bounded from below, and the vol-

ume constraint can be arbitrarily set to one. The dis-

cretized optimization problem can thus be written as,

minµ1,µ2,θ

: J (µ1,µ2,θ,U)= FTU

s.t. : K(µ1,µ2,θ)U = F

: vTm(µ1,µ2)− 1

ne∑e=1

ve ≤ 0

: 0 ≤ µ1,µ2

(2)

Where v is the vector containing the element vol-

umes ve. Stiffness matrix K is a function of µ1, µ2,

and θ, F is the load vector, and U describes the dis-

placement field. To avoid checkerboard patterns due to

the use of bi-linear finite elements, a standard density

filter with a filter radius of 1.5 element widths is ap-

plied on both fields µ1 and µ2 independently, (Bour-

din, 2001; Bruns and Tortorelli, 2001). Furthermore, as

is proposed in Groen and Sigmund (2017), small val-

ues of µ1 and µ2 need to be prevented to make a clear

distinction between regions consisting of material and

4 S. D. Larsen et al.

?

3L

L

F

??

3L

L

F

?

2L

L

F

L

?

4L

F3L

0.8L

(c) L-shaped domain (d) Michell cantilever with circular hole

(a) MBB-beam (b) MBB-beam with rectangular region of void

Fig. 2: Design domains and boundary conditions of examples used in this paper

regions which are void. To do so, the following material

interpolation scheme is used,

¯µi = µitanh(βη) + tanh(β(µi − η))

tanh(βη) + tanh(β(1− η))(3)

Where µi is the filtered width, and ¯µi is the physical

laminate width used to calculate the constitutive prop-

erties. By carefully choosing a continuation scheme for

the threshold parameter η, and the sharpness of the pro-

jection β, small widths between 0 and η can be banned

from the solution space with little effect on the perfor-

mance of the design. The parameters used for the con-

tinuation approach can be found in Figure 3, where the

legend shows the order of the scheme that is taken, us-

ing 50 iterations per step. Here the choice for η = 0.05,

means that no microstructures are allowed that contain

less than 5% of the average volume.

3 Mapping microstructures onto frame

structure

In recent works homogenization-based topologies have

been projected as smooth and continuous lattice struc-

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08µ1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

¯ µ1

β = 500, η = 0.00β = 100, η = 0.01β = 100, η = 0.02β = 100, η = 0.03β = 150, η = 0.04β = 150, η = 0.05β = 300, η = 0.05β = 1000, η = 0.05

Fig. 3: Interpolation scheme plotted for different values

of η and β, the order of the lines follows the continuation

approach

tures using two orthogonal layers.For each of the two or-

thogonal layers of the unit cell used in the homogenization-

based topology optimization, a mapping function is de-

termined that is locally aligned with the direction of

lamination. Using these two mapping functions, φ1 and

Optimal truss and frame design from projected homogenization-based topology optimization 5

φ2 respectively, a continuous sequence of unit cells can

be projected by means of cosine functions.

In this section, we present an alternative method to

create a near-optimal frame structure based on φ1 and

φ2, which subsequently can be used for further opti-

mization. The discussion on the derivation of the map-

ping functions will be kept limited, since this is not the

main goal of this work. For a detailed derivation the

reader is referred to Groen and Sigmund (2017).

3.1 Mapping a periodic composite shape

Mapping functions φ1 and φ2 can be obtained indepen-

dently of each other, using a similar approach, therefore

we restrict ourselves to the derivation of φ1. A suitable

parameterization of φ1 has to fulfill two requirements:

1. φ1 should be constant in the direction of lamination

in non-void domains, such that the frame structure

can be described as contour lines.

2. The spacing between the contour lines of φ1 in these

non-void domains, should be as regular as possible

without violating the first requirement.

These requirements are not equally weighted through-

out domain Ω. As shown in Figure 1, a part of the

homogenization-based topology does not consist of ma-

terial, hence the mapping functions do not have to be

accurate here. Furthermore, requirement 1 is too strict

in regions where the angle field changes rapidly, e.g.

at the Dirichlet BC’s in the cantilever beam example.

Therefore domain Ω is split in three subdomains, a

smooth lattice domain Ωl, a void domain Ωv, and a

domain Ωθ in which the angle field is rapidly changing.

These domains are defined as,

x ∈

Ωv if ¯µ1(x), ¯µ2(x) < η

Ωθ if ¯µ1(x), ¯µ2(x) ≥ η and ∇θ(x) > γθ

Ωl if ¯µ1(x), ¯µ2(x) ≥ η and ∇θ(x) < γθ

(4)

where, γθ is a threshold that dictates whether the an-

gular field is rapidly changing or not.

γθ =π

4

1

hc(5)

Here hc is the element length used in the homogenization-

based topology optimization. Due to the selected thresh-

old, Ωθ contains the parts of the domain where the

angle field is close to singular. Using these different do-

mains, we can solve for mapping function φ1 by means

of a spatially weighted constrained least-squares mini-

mization problem.

minφ1(x)

: I(φ1(x)) =1

2

∫Ω

α1(x) ‖∇φ1(x)− e1(x)‖2 dΩ

s.t. : α2(x)∇φ1(x) · e2(x) = 0

(6)

where,

α1(x) =

0.01 if x ∈ Ωv0.1 if x ∈ Ωθ1 if x ∈ Ωl

, α2(x) =

0 if x ∈ Ωv0 if x ∈ Ωθ1 if x ∈ Ωl

(7)

Furthermore, unit vectors e1, e2 depend on the local

directions of lamination θ,

e1 =

[−sin(θ)

cos(θ)

], e2 =

[cos(θ)

sin(θ)

](8)

It can be seen that the constraint, which dictates exact

angular enforcement, is only active in Ωl, i.e. the part

of the domain where the lattice is smooth. Experiments

have shown that a high gradient in angle field in com-

bination with angular enforcement results in a severely

distorted lattice spacing, hence the calculation of φ1 in

Ωθ is relaxed. Since φ1 does not have to be accurate

in Ωv, its calculation is heavily relaxed to allow for the

best approximation in other parts of Ω.

Furthermore, it has to be noted that the princi-

pal stress directions used to calculate θ are rotation-

ally symmetric, hence there may be jumps of size π

in angle field θ. These jumps are identified using con-

nected component labeling and aligned consistently as

suggested in Groen and Sigmund (2017), to allow for a

smooth projection using Equation 6.

The mapping functions for the Michell cantilever are

shown in Figure 4(a). The microstructure is optimized

on a coarse mesh of 80×40 elements. Afterwards, map-

ping functions φ1 and φ2 are calculated on a six times

finer mesh of 480 × 240 elements, This is done to get

smooth and accurate values for φ1 and φ2 yet still at

low computational cost.

3.2 Extraction of nodes and connectivity

The contour lines of mapping functions φ1 and φ2, shown

in Figure 4(b) resemble a frame-like structure. Using

standard MATLAB functions (e.g. contour), these con-

tour lines can be extracted, from which nodal positions

and connectivity of the initial frame structure are es-

tablished. To obtain highly accurate locations of the

6 S. D. Larsen et al.

(a) Mapping functions φ1 and φ2 (b) Corresponding contour lines

(c) Elements and nodes for ε = 20 hf (d) Elements and nodes for ε = 50 hf

Fig. 4: The nodes and element-connectivity extracted from the mapping functions for the Michell cantilever problem

contour lines, mapping functions φ1 and φ2 are inter-

polated on a fine mesh of 1600 × 800 elements, i.e. 20

times finer than the mesh used for homogenization-

based topology optimization.

For simplicity, we choose to draw a contour line

when mapping function φi takes a whole value. To influ-

ence the number of contour lines, φi is multiplied with

periodicity scaling parameter Pi, which is based on a

user-defined average length of a frame member ε.

Pi =1

ε

∫(Ω\Ωv) d(Ω \Ωv)∫

(Ω\Ωv) ‖∇φi(x)‖2 d(Ω \Ωv)(9)

Where the integrals scale mapping functions φi w.r.t.

their average spacing. It can easily be seen that a large

value of ε results in a small number of nodes and ele-

ments, while a small value results in a detailed frame

structure. Finally, to make sure that a contour line

passes through a specific point, e.g. the load-node, func-

tions φ1 and φ2 can be shifted, before the contour-lines

are extracted. An overview of the nodes and element

connectivity for the Michell cantilever can be seen in

Figure 4, here figures (c) and (d) show the nodes and

connectivity for ε = 20 hf , and ε = 50 hf respectively.

3.3 Mapping of material distribution to element areas

Relative areas are assigned to the frame elements based

on the values of ¯µ1, and ¯µ2 from the homogenization-

based topology optimization. To transfer the contin-

uous material distribution to the discrete elements, apolygon is drawn around each element, which describes

the area in the continuum domain the element cov-

ers. An example of such a polygon can be seen in Fig-

ure 5(a). These polygons are obtained using the spacing

of the contour lines, wi, which can be approximated lo-

cally using mapping function φi.

wi(x) =1

‖∇φi(x)‖2(10)

For each node of an element this spacing is calculated,

and by taking a step orthogonal to the element with

a stepsize of half the spacing the corresponding poly-

gon can be drawn. By integrating the values of ¯µ1 or¯µ2 in each of the polygons and dividing by the element

length, a near-optimal initial area distribution of the

frame structure is obtained based on the continuum so-

lution, see e.g. Figure 5(b). Finally, we can identify the

threshold area Aη, which is the area that an average

sized frame element (i.e. with length ε) should have if

Optimal truss and frame design from projected homogenization-based topology optimization 7

¯µi = η.

Aη = ηε (11)

Bars that have an area smaller than this value, occur

when a part of the polygon, used for integrating the

volume, is within Ωv. Hence, elements with a mapped

area that is smaller than Aη are removed, as can be

seen in Figure 5(c).

(a) Polygon around element to integrate volume

(b) Material mapped to elements

(c) Removal of elements with area smaller than Aη

Fig. 5: Procedure for mapping material from the con-

tinuum solution of homogenization-based topology op-

timization to discrete frame elements

3.4 Assigning boundary conditions

A distinction is made between how two different types

of boundary conditions (BC’s) are applied to the frame

structure. Most nodes to which a BC needs to be

assigned (in this work referred to as standard-BC’s)

can be identified easily, e.g. by finding the intersec-

tion with a line along which the BC’s have to be ap-

plied. However, the analytical solutions for Michell’s

problem of least-weight trusses (Hemp, 1973; Lewinski

et al., 1994a,b; Lewinski and Rozvany, 2008), also con-

sists of boundary conditions that can be interpreted

as source points from which multiple elements origi-

nate. These so-called fan-BC’s can be found for exam-

ple at the nodal Dirichlet BC’s in the analytical solu-

tion of the Michell cantilever example (Lewinski et al.,

1994b). Careful inspection of the contour lines obtained

for this example (Figure 4(b)) reveals that multiple

contour lines point to the location where these BC’s

have to be applied. To allow for these fan-BC’s in the

initial structure, all elements and nodes within radius

RBC = 1/8 L of both BC’s are pulled exactly into the

boundary nodes, as is shown in Figure 6.

Fig. 6: Operation that pulls in all nodes and elements

within RBC = 100 hf to generate a fan-BC

We can automatically identify if a boundary condi-

tion needs to be a standard-BC or a fan-BC. Therefore,

we look at orientation field θ, inside radius RBC of the

boundary nodes. At locations where a fan-BC needs to

be inserted, the angular field is rapidly changing, hence

we can simply check if points close to a nodal BC are

inside Ωθ.

3.5 Preparation of initial frame structure

To clean up and improve the stability of the initial

structure, normal nodes (i.e. non-boundary conditions)

connected to only one or two elements are removed. In

the case of a node connected to a single element, the

element cannot carry any load, hence both node and el-

ement are removed. Furthermore, connections between

two elements are unstable and do not exist in the solu-

tion space for Michell’s problem of least-weight trusses.

Therefore the corresponding elements are merged into a

8 S. D. Larsen et al.

single element, while the node is removed. In some sit-

uations this operation will result in crossing elements.

If this happens the largest element in a crossing is re-

moved. Finally, it has to be mentioned that all opera-

tions that modify the structure, e.g. assigning BC’s, are

made volume preserving. Hence, the relative material

distribution throughout the domain remains as close

to the material distribution of the continuum model as

possible.

The initial frame structure for the Michell cantilever

obtained using the approach described above, for ε =

50 hf can be seen in Figure 7(a). Similarly, the initial

structures for the MBB-beam and the Michell cantilever

with circular support, both for ε = 50 hf , can be seen

in Figures 7(b) and (c) respectively.

3.6 Void regions inside design domain

The mapping procedure described above can easily be

extended to take specified void regions in the design

domain into account. However, special care needs to be

taken at the corners of these void regions, since the ho-

mogenization based microstructures are either oriented

parallel to the void region, or oriented to go exactly

through the corner point of a void region. This highly

optimal use of the available design space can result in

mapped elements that cross the void domain. Further-

more, it should be noted that corners of a void region

can also be a source of a fan similar to the fan-BC’s. As

discussed by Lewinski and Rozvany (2008) the optimal

solution for the L-shaped domain, shown in Figure 2(c),

consists of a fan at the corner of the void domain.

To accommodate for these fan corner nodes, a check

is performed for each corner of a specified void domain.

If points within RBC of these corners are inside Ωθ, a

fan is created in the exact same manner as a fan-BC.

At a corner of the void domain, which is not a fan, and

at which an element crosses, a node is inserted and the

crossing element is split in two elements.

The projected frame structure for the L-shaped do-

main can be seen in Figure 8(a), the initial frame struc-

ture for the MBB-beam with a rectangular region of

void is shown in Figure 8(b), where for both structures

ε = 50 hf .

4 Post size- and geometry-optimization

In this section we propose a frame optimization scheme

that ensures solutions close to the solutions of Michell’s

problems of least-weight trusses, hence with a negligi-

ble bending contribution. Furthermore, we propose a

(a) Michell cantilever

(b) MBB-beam

(c) Michell cantilever with circular support

Fig. 7: Initial structures extracted from

homogenization-based topology optimization for

ε = 50 hf

strategy to avoid: 1) elements that are very thin, 2) el-

ements that are very short, and 3) elements that are

parallel and partially overlap, since each of these three

cases result in a singular stiffness matrix when modeled

as truss.

4.1 Motivation for frame analysis

It is well-known that solutions of Michell’s problem of

least-weight trusses are in the space of statically de-

terminate structures, (Pedersen, 1969). However, it is

Optimal truss and frame design from projected homogenization-based topology optimization 9

(a) L-shaped domain

(b) MBB-beam with a rectangular region of void

Fig. 8: Initial structures extracted from

homogenization-based topology optimization for

ε = 50 hf

not possible to use truss elements to assess the perfor-

mance of the mapped structures. Small misalignments

close to boundary conditions, e.g. at the symmetry con-

ditions for the MBB-beam example, may result in inde-

terminate structures. Nevertheless, a post-optimization

scheme is required, such that the projected structures

converges towards solutions of Michell’s problem of least-

weight trusses. To do so, frame elements are used that,

contrary to truss elements, not only carry axial loads

but also have bending stiffness.

For such an analysis appropriate relations between

the domain length L and element areas have to be

chosen, such that the bending stiffness does not be-

come dominant. A circular cross-section is chosen for

the frame elements, hence for given element i the re-

lation between axial (ka) and bending (kb) stiffness is

given by,

kakb∝ l2iAi

(12)

where li is the length of the element and Ai the corre-

sponding area.

The design vector for geometry optimization xn holds

the coordinates of the nodal positions. To get a well con-

ditioned optimization problem the length of the domain

is scaled such that xn ∈ [0, 10]. The vector of element

areas A is obtained using the design vector for size opti-

mization xe, and the maximum allowable element area

Amax.

A(xe) = xeAmax ∀ xe ∈ [0, 1[ (13)

Here xe is based on the relative material distribution

in the mapped elements. The relative values in xe are

scaled down far enough such that the upper bound of

1 never becomes active to prevent that A > Amax.

Therefore, the relation between bending and axial stiff-

ness is only controlled by choosing an appropriate value

for Amax. Hence, Amax is chosen differently for each

optimization example such that the relation between

the mean values of kakb

(i.e. kakb

) is always exactly the

same at the start of the post-optimization scheme, e.g.kakb

= 100.

To converge towards a design that is purely loaded

in axial direction, the relative importance of the bend-

ing stiffness is slowly decreased using a continuation

scheme. In this scheme Amax and the volume constraint

V ∗ are lowered by 12.5 % for every 10 iterations. This

does not have an effect on the relative distribution of

axial loads; however, it does make it uneconomical to

have elements that are not purely loaded in axial di-

rection. The continuation scheme start after the first

100 iterations and is continued until Amax is less than

2.5% of its initial value. At this point the contribution

of the bending stiffness to the strain energy is negligi-

ble, as will be discussed in more detail in Section 5. The

choice for the steps used in the continuation scheme re-

sult from a trade-off between the performance of the

design and computational cost. For smaller steps, sig-

nificantly more iterations are required to optimize the

design, resulting in a slightly better objective. Similarly,

a larger stepwise reduction in bending stiffness, means

that the algorithm converges more quickly; however, re-

sulting in a reduced performance.

4.2 Optimization scheme

The frame optimization problem is solved to minimize

compliance Jf , subject to a volume constraint V ∗, which

is the amount of material in the frame members at the

first iteration. The optimization problem is solved in

nested form using a gradient-based optimization scheme,

where we use the Method of Moving Asymptotes (MMA)

10 S. D. Larsen et al.

to update the design variables (Svanberg, 1987). The

corresponding optimization problem can be written as,

minxn,xe

: Jf (xn,xe,U) = FTU

s.t. : K(xn,xe)U = F

:

∑nei=1 li(xn)Ai(xe)

V ∗− 1 ≤ 0

: xn,l ≤ xn ≤ xn,u

: xe,l ≤ xe ≤ xe,u

(14)

Where xe,l, xe,u, xn,l and xn,u are the lower and up-

per bounds for the size and geometry design vectors

respectively. For each design iteration lower and upper

bounds on the design variables are adaptively selected.

These bounds are chosen such that the design changes

gradually, furthermore, the bounds on xn make sure

that elements will not cross each other, or move into

the specified void domains.

4.3 Removal of thin elements

During the optimization, values in xe can become close

to 0, hence these elements contain almost no material.

Typically, these are prevented by using a lower bound

on the areas; however, this adds artificial stiffness to the

structures, and can also prevent structures from becom-

ing statically determinate. Therefore, elements smaller

than a selected threshold will be removed from the so-

lution space. This threshold is based on the value of

Aη used in the mapping procedure, but scaled with

the same factor used to obtain xe. Furthermore, the

threshold is consistently updated during the continua-

tion scheme.

By removing thin elements from the frame mesh,

normal nodes (i.e non-BC’s) can be connected to only

one or two elements, making the structure unstable. To

avoid this undesired effect, the exact same procedure is

applied as discussed in Section 3.5.

4.4 Merging of close nodes

It is well-known that for some elements the nodes move

towards the same point, making the corresponding el-

ement length zero. This effect, sometimes referred to

as melting nodes (Achtziger, 2007), will cause a singu-

larity in the stiffness matrix and is therefore undesired.

To take this into account, the nodes of elements shorter

than a selected threshold lshort will be merged, remov-

ing a node from the solution space in an approach simi-

lar to (He and Gilbert, 2015). The value for the thresh-

old lshort is selected to be one fifth of the average size

of the projected element ε, scaled with the same scaling

factor used to obtain xn.

By merging nodes, it is possible that non-unique el-

ements exist between two nodes. To take this undesired

effect into account, one of the corresponding elements is

removed, and the volume of both elements is contained

by the remaining element.

4.5 Merging of parallel and partially overlapping

elements

It is possible that during post-optimization two or more

elements, located along boundaries of the design do-

main are partially overlapping and parallel. This sit-

uation can be observed at the lower boundary of the

MBB-beam example with a void, shown in Figure 9.

Here, there is an element between node 1 and node 2,

an element between node 2 and node 3, and an element

between node 1 and node 3.

Although all three elements are unique, it is un-

physical that the elements overlap, furthermore, this

can have an undesired effect on the condition-number

of the stiffness matrix. To remedy this, the longest el-

ement is split into two elements. In this case the two

smaller elements already exist, and the volume of the

longest elements is transferred to the smaller elements

consistently.

5 Numerical examples

An overview of all parameters used in this work can be

found in Table 1. The horizontal lines are used to show

a division in parameters used in: 1) homogenization-

based topology optimization, 2) calculation of mapping

functions, 3) post-optimization. Please note that we use

ε = 50 hf , unless otherwise stated. In the following, we

demonstrate the suggested procedure on a number of

examples and compare with analytical solutions when

available.

5.1 Michell cantilever

The near-optimal initial structure for the Michell can-

tilever, shown in Figure 7(a), has been optimized us-

ing the presented post-optimization scheme. The result,

shown in Figure 10, can be modeled as a truss, i.e. it is

statically determinate.

To assess the performance of the optimized design,

one can look at the non-dimensional mass when evalu-

ated as truss M t, which for a Michell cantilever can be

Optimal truss and frame design from projected homogenization-based topology optimization 11

1 2 3

Fig. 9: MBB-beam with a rectangular region of void, to demonstrate that elements can be parallel and partially

overlap. There is an element between node 1 and node 2, an element between node 2 and node 3, and an element

between node 1 and node 3

Table 1: Parameters used in the numerical experiments

Parameter Definition ValueE Young’s modulus of material in continuum model 1Emin Young’s modulus of background material 0.005 Eν Poisson’s ratio for isotropic background material 0.3rmin Filter radius used in continuum topology optimization 1.5 hcη Minimum feature size per layer in the microstructure 0.05

γθ Threshold value that determines whether angular field is rapidly changing π4

1hc

α1 Spatially variant parameter to relax the objective of projection [0.01, 0.1 , 1]α2 Spatially variant parameter to relax the constraint of projection [0, 1]ε average length of the frame member 50 hfRBC Radius to search for a fan-BC, and to pull in all nodes to BC 1/8 Llshort Short element length which dictates when two nodes are merged 0.2 εkakb

Starting relation between the mean axial and bending stiffness 100Amax,endAmax,start

Measure for reduction in bending stiffness 0.025

Fig. 10: Optimized structure for the Michell cantilever,

using initial structure for ε = 50 hf

calculated as (Rozvany, 1998; Bendsøe et al., 1994),

M t =

√MEJtFL

(15)

Here M is the volume of the final structure, and Jtthe compliance when modeled as a truss structure. The

non-dimensional mass for the optimized Michell can-

tilever is 7.0391, which is close to the optimal value,

Mopt = 7.0247, found in a table in Graczykowski and

Lewinski (2010). This means that the non-dimensional

mass of the optimized structure is just 0.204% higher

than the analytical optimum. Furthermore, the compli-

ance when the structure is modeled as a truss Jt, is

almost identical to the compliance modeled as a frame

structure Jf = 7.0390, used in the post-optimization

scheme. It is possible to identify a measure (fb) of the

total contribution of the bending stiffness on Jf .

fb =Jf −UT

t FtJf

× 100% (16)

Here Ut and Ft correspond to the displacement in-

dices of the solution and load vector respectively, i.e.

excluding indices corresponding to rotation. The bend-

ing contribution for the optimized Michell cantilever

is 0.00045%. A plot of this contribution for each iter-

ation of the post-optimization scheme can be seen in

Figure 11.

Due to the continuation scheme, the bending con-

tribution is lowered every 10 iterations after the 100th

iteration. However, it is more interesting to see that

the bending contribution is drastically reduced in the

first few iterations. The reason is that tiny misalign-

ments of nodal positions close to the boundary condi-

tions severely deteriorate the performance of the struc-

ture (e.g. singular matrix when modeled as truss). When

modeled as a frame structure, the initial bending stiff-

ness provides stability; however, the performance is im-

proved when these nodes are better aligned, hence the

12 S. D. Larsen et al.

Fig. 11: Contribution of the bending stiffness to the

overall compliance for the Michell cantilever

contribution of the bending to the compliance is re-

duced.

The optimization is performed for the Michell can-

tilever using three different levels of detail of the initial

structure, i.e. ε = 20 hf , ε = 50 hf , and ε = 100 hf .

The optimized structures for ε = 20 hf , and ε = 100 hfcan be seen in Figure 12(a) and (b) respectively.

(a) ε = 20 hf

(b) ε = 100 hf

Fig. 12: Optimized structures for the Michell cantilever

The corresponding size of the fine mesh on which the

nodes and elements are obtained, the number of nodes

Nn, number of elements Ne, the performance measured

in non-dimensional masses (Mf and Mopt), the error ξ

between Mf and the analytical solution Mopt and the

contribution of the bending stiffness fb can be found

in Table 2. As expected, a finer initial structure results

in a better performing design. Furthermore, the time

to do the homogenization-based topology optimization

on the coarse mesh Tc, the time to obtain the initial

structure Tφ, the time to do the post-optimization Tf ,

and the total time Ttot are shown. Here it has to be

noted that all experiments are performed using a single

processor MATLAB code on a standard PC. Hence,

large potential for further time reduction exists.

It can be seen that a more detailed structure comes

at a larger computational cost, which is dominated by

the sensitivity analysis for the post-optimization scheme.

Furthermore, the increased level of detail means an in-

crease in computational cost to obtain the initial mesh,

since more contour lines of mapping function φ1 and

φ2 need to be considered. Nevertheless, the increase in

computational cost is significantly smaller compared to

growth methods, where the time to insert a new mem-

ber scales exponentially.

5.2 Stability of optimized results

The far majority of the optimized structures for the

other examples is unstable when modeled as a truss. To

explain this we can take another look at Figure 9, where

the bottom of the MBB-beam with a rectangular void is

shown. In this example, a condition for stability when

modeled as truss, is that node 1 and node 2, should

have the exact same value for their y-coordinate. Even

the slightest misalignment (e.g. 10−6) will result in an

unstable structure since node 1, is only supported in

the x-direction.

When modeled as frame, even the smallest bending

contribution, will prevent such an instability. Hence,

for the post-optimized MBB-beam shown in Figure 13,

the element that is connected to the middle node of

the symmetry boundary is nearly horizontal; however

not exactly. Unfortunately, this statical indeterminacy

means that we cannot assess the performance as a truss,

and hence an exact comparison between the optimal

value for the non-dimensional mass, Mopt = 14.0937

and M t is not possible. Nevertheless, we argue that the

bending contribution is sufficiently small, fb = 0.0055%

to allow for a comparison between Mopt and the non-

dimensional weight calculated with the compliance from

the frame model, Mf = 14.1878.

To demonstrate this we decrease Amax by different

orders of magnitude, such that the bending stiffness of

the node at the symmetry boundary is reduced. As can

Optimal truss and frame design from projected homogenization-based topology optimization 13

Table 2: Performance and computational cost of the near optimal truss and frame structures. Where example 1) is

the Michell cantilever, 2) the Michell cantilever with circular support, 3) the MBB-beam, 4) the MBB-beam with

void, and 5) the L-shaped domain

Ex. Proj. mesh ε Nn Ne Mf Mopt ξ fb Tc Tφ Tf Ttot1 1600 × 800 20 hf 453 900 7.0327 7.0247 0.114% 0.00032% 96.0 s 104.2 s 339.9 s 540.0 s1 1600 × 800 50 hf 103 202 7.0392 7.0247 0.206% 0.00045% 96.0 s 22.9 s 67.5 s 186.4 s1 1600 × 800 100 hf 32 60 7.0545 7.0247 0.422% 0.00059% 96.0 s 18.4 s 29.3 s 143.7 s2 1600 × 1200 50 hf 380 710 2.1248 2.1401 −0.720% 0.0173% 136.3 s 47.0 s 288.4 s 487.5 s2 1600 × 1200 100 hf 122 212 2.1474 2.1401 0.340% 0.0021% 136.3 s 30.5 s 72.7 s 255.3 s3 2520 × 840 50 hf 106 208 14.1878 14.0937 0.663% 0.0055% 152.1 s 45.1 s 94.8 s 292.0 s3 2520 × 840 100 hf 40 74 14.2675 14.0937 1.218% 0.0085% 152.1 s 32.6 s 43.8 s 228.4 s4 2520 × 840 50 hf 40 67 14.5616 - - 0.0029% 146.6 s 48.5 s 65.8 s 260.9 s4 2520 × 840 100 hf 19 28 14.6425 - - 0.0031% 146.6 s 43.3 s 37.8 s 227.7 s5 1600 × 1600 50 hf 26 46 9.3004 9.283 0.187% 0.0074% 178.4 s 53.3 s 58.2 s 289.8 s5 1600 × 1600 100 hf 15 24 9.3284 9.283 0.487% 0.0044% 178.4 s 45.8 s 37.0 s 261.2 s

Fig. 13: Optimized structure for the MBB-beam, using

initial structure for ε = 50 hf

be seen in Table 3, this will lead to an increase in non-

dimensional mass of the frame model (Mf ). However,

the contribution of the purely axial stiffness M(UTt Ft)

remains almost perfectly intact. The reason for this is

that the node at the symmetry boundary is the only

node that causes an instability. Only this node is af-

fected by a decrease in bending stiffness, hence the in-

crease in strain energy in the system is purely due to

the near horizontal element being stretched. This in-

crease in energy has a negligible effect on the energy in

the rest of the system, even for low values of Amax, and

therefore we argue that Mf can be used to compare

with the optimal solution Mopt.

Table 3: Non-dimensional mass of the frame struc-

ture Mf , and purely axial contribution of the non-

dimensional mass M(UTt Ft) when the area is scaled

down

Area scaling Mf M(UTt Ft) fbAmax 14.1878 14.1870 0.0055%

10−1 Amax 14.1890 14.1890 0.0050%10−2 Amax 14.1940 14.1890 0.0340%10−3 Amax 14.2370 14.1890 0.3360%10−4 Amax 14.6600 14.1890 3.2120%10−5 Amax 18.3610 14.1900 22.7140%10−6 Amax 39.4440 14.3250 63.6830%

5.3 Discussion of results

The post-optimized structures for the MBB-beam, Michell

cantilever with circular support, MBB-beam with rect-

angular void and L-shaped domain, all for ε = 50 hfcan be seen in Figure 13, Figure 14(a), (b) and (c) re-

spectively. While the corresponding performance and

the different times can be seen in Table 2. It is interest-

ing to see that the optimized structures perform very

close to the optimal solution, at a negligible bending

contribution. However, the performance of the MBB-

beam with rectangular void cannot be compared to an

analytical solution, since this solution is not known.

Furthermore, it is interesting to see that the non-

dimensional mass for the Michell cantilever with circu-

lar support is lower than the analytical optimum. This

does not come from the fact that it is modeled as a

frame, nor does it come from the analytical solution

being wrong. The simple reason is that the boundary

of the extracted structure is not perfectly circular as can

be seen in Figure 14(a). This jagged boundary has its

origin in the coarse-scale homogenization-based topol-

ogy optimization model, where the circular boundary

is approximated by coarse square elements. In our al-

gorithm, this is the boundary that is transferred down

from the coarse-scale topology optimization model to

the frame model, hence the difference with the analyt-

ical optimum.

Besides the fact that the final structures perform

well, it has to mentioned that the total procedure comes

at a relatively low computational cost, i.e. all exam-

ples have been obtained within 10 minutes using a sin-

gle processor MATLAB code on a standard PC. The

homogenization-based topology optimization can be done

in a couple of minutes, where the difference in time be-

tween the examples comes from different mesh sizes.

E.g. the Michell cantilever is optimized on a mesh of

80 × 40 elements, while the L-shaped domain is opti-

14 S. D. Larsen et al.

(a) Michell cantilever with circular support

(b) MBB-beam with rectangular void

(c) L-shaped domain

Fig. 14: Optimized structures extracted from

homogenization-based topology optimization for

ε = 50 hf

mized on a mesh consisting of 80 × 80 elements. The

extraction time for the initial structure does increase

when the average spacing ε is decreased. However, this

increase in computational cost is more or less quadrat-

ically related to 1/ε, which is a significant advantage

over growth methods that scale exponentially when fine

designs are considered.

6 Conclusion

An approach to obtain near-optimal frame struc-

tures has been presented, where the discrete struc-

tures are based on the solution of a homogenization-

based topology optimization model. The coarse-scale

homogenization-based continuum solution is used to

create a close to optimal initial structure, which is ob-

tained by solving for two mapping functions and using

their corresponding contour lines. Furthermore, accu-

rate integration of the continuum solution allows for a

good starting guess for the element areas. Afterwards,

these initial structures are optimized using a frame op-

timization code, to avoid the problem of a singular ma-

trix when modeled as truss. To make sure that the final

structures are close to the known solutions of Michell’s

problem of least–weight trusses, we gradually reduce

the bending stiffness such that the final structures are

only loaded in axial direction.

Based on numerical experiments, we can conclude

that the presented approach produces near-optimal frame

structures at a relatively low computational cost. This

promising performance paves the way for extending the

methodology to multiple load problems. In these prob-

lems, the optimal continuum solution is in the space of

rank-3 microstructures, compared to the orthogonal mi-

crostructures used in the current approach. The exten-

sion to 3-dimensions is also possible, based on orthog-

onal projection of (sub-optimal) truss-like microstruc-

tures.

Acknowledgements The authors acknowledge the supportof the Villum Fonden through the Villum investigator projectInnoTop. The authors would also like to thank Andreas Bærentzenand Niels Aage for valuable discussions during the prepara-tion of the work. Finally, the authors wish to thank KristerSvanberg for providing the MATLAB MMA code.

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